How should we test the hypothesis?

You are interested in the presence of lane bias toward higher lanes, presumably due to a slight current in the pool. A natural null hypothesis to test, then, is that the mean fractional improvement going from low to high lane numbers is zero. Which of the following is a good way to simulate this null hypothesis?

As a reminder, the arrays swimtime_low_lanes and swimtime_high_lanes contain the swim times for lanes 1-3 and 6-8, respectively, and we define the fractional improvement as f = (swimtime_low_lanes - swimtime_high_lanes) / swimtime_low_lanes.

Possible Answers

Randomly swap swimtime_low_lanes[i] and swimtime_high_lanes[i] with probability 0.5. From these randomly swapped arrays, compute the fractional improvement. The test statistic is the mean of this new f array.
Scramble the entries in the swimtime_high_lanes, and recompute f from the scrambled array and the swimtime_low_lanes array. The test statistic is the mean of this new f array.
Shift the swimtime_low_lanes and swimtime_high_lanes arrays by adding a constant value to each so that the shifted arrays have the same mean. Compute the fractional improvement, f_shift, from these shifted arrays. Then, take a bootstrap replicate of the mean from f_shift.
Subtract the mean of f from f to generate f_shift. Then, take bootstrap replicate of the mean from this f_shift.
Either (3) or (4) will work; they are equivalent.

Exercise

How should we test the hypothesis?

Instructions

Possible Answers